Integrand size = 24, antiderivative size = 70 \[ \int \frac {x^3 \sqrt {c+d x^4}}{a+b x^4} \, dx=\frac {\sqrt {c+d x^4}}{2 b}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{2 b^{3/2}} \] Output:
1/2*(d*x^4+c)^(1/2)/b-1/2*(-a*d+b*c)^(1/2)*arctanh(b^(1/2)*(d*x^4+c)^(1/2) /(-a*d+b*c)^(1/2))/b^(3/2)
Time = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.99 \[ \int \frac {x^3 \sqrt {c+d x^4}}{a+b x^4} \, dx=\frac {1}{2} \left (\frac {\sqrt {c+d x^4}}{b}-\frac {\sqrt {-b c+a d} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {-b c+a d}}\right )}{b^{3/2}}\right ) \] Input:
Integrate[(x^3*Sqrt[c + d*x^4])/(a + b*x^4),x]
Output:
(Sqrt[c + d*x^4]/b - (Sqrt[-(b*c) + a*d]*ArcTan[(Sqrt[b]*Sqrt[c + d*x^4])/ Sqrt[-(b*c) + a*d]])/b^(3/2))/2
Time = 0.35 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {946, 60, 73, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^3 \sqrt {c+d x^4}}{a+b x^4} \, dx\) |
\(\Big \downarrow \) 946 |
\(\displaystyle \frac {1}{4} \int \frac {\sqrt {d x^4+c}}{b x^4+a}dx^4\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{4} \left (\frac {(b c-a d) \int \frac {1}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^4}{b}+\frac {2 \sqrt {c+d x^4}}{b}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{4} \left (\frac {2 (b c-a d) \int \frac {1}{\frac {b x^8}{d}+a-\frac {b c}{d}}d\sqrt {d x^4+c}}{b d}+\frac {2 \sqrt {c+d x^4}}{b}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{4} \left (\frac {2 \sqrt {c+d x^4}}{b}-\frac {2 \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{b^{3/2}}\right )\) |
Input:
Int[(x^3*Sqrt[c + d*x^4])/(a + b*x^4),x]
Output:
((2*Sqrt[c + d*x^4])/b - (2*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^ 4])/Sqrt[b*c - a*d]])/b^(3/2))/4
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m - n + 1, 0]
Time = 0.15 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.89
method | result | size |
pseudoelliptic | \(\frac {\sqrt {d \,x^{4}+c}-\frac {\left (a d -c b \right ) \arctan \left (\frac {\sqrt {d \,x^{4}+c}\, b}{\sqrt {\left (a d -c b \right ) b}}\right )}{\sqrt {\left (a d -c b \right ) b}}}{2 b}\) | \(62\) |
risch | \(\frac {\sqrt {d \,x^{4}+c}}{2 b}-\frac {\left (a d -c b \right ) \left (-\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 b \sqrt {-\frac {a d -c b}{b}}}-\frac {\ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 b \sqrt {-\frac {a d -c b}{b}}}\right )}{b}\) | \(344\) |
default | \(\frac {\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}+\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {\frac {d \sqrt {-a b}}{b}+\left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{\sqrt {d}}+\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}\right )}{b}+\frac {\left (a d -c b \right ) \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -c b}{b}}}}{4 b}+\frac {\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}-\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {-\frac {d \sqrt {-a b}}{b}+\left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{\sqrt {d}}+\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}\right )}{b}+\frac {\left (a d -c b \right ) \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -c b}{b}}}}{4 b}\) | \(683\) |
elliptic | \(\frac {\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}+\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {\frac {d \sqrt {-a b}}{b}+\left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{\sqrt {d}}+\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}\right )}{b}+\frac {\left (a d -c b \right ) \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -c b}{b}}}}{4 b}+\frac {\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}-\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {-\frac {d \sqrt {-a b}}{b}+\left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{\sqrt {d}}+\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}\right )}{b}+\frac {\left (a d -c b \right ) \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -c b}{b}}}}{4 b}\) | \(683\) |
Input:
int(x^3*(d*x^4+c)^(1/2)/(b*x^4+a),x,method=_RETURNVERBOSE)
Output:
1/2/b*((d*x^4+c)^(1/2)-(a*d-b*c)*arctan((d*x^4+c)^(1/2)*b/((a*d-b*c)*b)^(1 /2))/((a*d-b*c)*b)^(1/2))
Time = 0.10 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.23 \[ \int \frac {x^3 \sqrt {c+d x^4}}{a+b x^4} \, dx=\left [\frac {\sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x^{4} + 2 \, b c - a d - 2 \, \sqrt {d x^{4} + c} b \sqrt {\frac {b c - a d}{b}}}{b x^{4} + a}\right ) + 2 \, \sqrt {d x^{4} + c}}{4 \, b}, -\frac {\sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x^{4} + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) - \sqrt {d x^{4} + c}}{2 \, b}\right ] \] Input:
integrate(x^3*(d*x^4+c)^(1/2)/(b*x^4+a),x, algorithm="fricas")
Output:
[1/4*(sqrt((b*c - a*d)/b)*log((b*d*x^4 + 2*b*c - a*d - 2*sqrt(d*x^4 + c)*b *sqrt((b*c - a*d)/b))/(b*x^4 + a)) + 2*sqrt(d*x^4 + c))/b, -1/2*(sqrt(-(b* c - a*d)/b)*arctan(-sqrt(d*x^4 + c)*b*sqrt(-(b*c - a*d)/b)/(b*c - a*d)) - sqrt(d*x^4 + c))/b]
Time = 4.35 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.31 \[ \int \frac {x^3 \sqrt {c+d x^4}}{a+b x^4} \, dx=\begin {cases} \frac {2 \left (\frac {d \sqrt {c + d x^{4}}}{4 b} - \frac {d \left (a d - b c\right ) \operatorname {atan}{\left (\frac {\sqrt {c + d x^{4}}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{4 b^{2} \sqrt {\frac {a d - b c}{b}}}\right )}{d} & \text {for}\: d \neq 0 \\\sqrt {c} \left (\begin {cases} \frac {x^{4}}{4 a} & \text {for}\: b = 0 \\\frac {\log {\left (4 a + 4 b x^{4} \right )}}{4 b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \] Input:
integrate(x**3*(d*x**4+c)**(1/2)/(b*x**4+a),x)
Output:
Piecewise((2*(d*sqrt(c + d*x**4)/(4*b) - d*(a*d - b*c)*atan(sqrt(c + d*x** 4)/sqrt((a*d - b*c)/b))/(4*b**2*sqrt((a*d - b*c)/b)))/d, Ne(d, 0)), (sqrt( c)*Piecewise((x**4/(4*a), Eq(b, 0)), (log(4*a + 4*b*x**4)/(4*b), True)), T rue))
Exception generated. \[ \int \frac {x^3 \sqrt {c+d x^4}}{a+b x^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^3*(d*x^4+c)^(1/2)/(b*x^4+a),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Time = 0.12 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.94 \[ \int \frac {x^3 \sqrt {c+d x^4}}{a+b x^4} \, dx=\frac {{\left (b c - a d\right )} \arctan \left (\frac {\sqrt {d x^{4} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, \sqrt {-b^{2} c + a b d} b} + \frac {\sqrt {d x^{4} + c}}{2 \, b} \] Input:
integrate(x^3*(d*x^4+c)^(1/2)/(b*x^4+a),x, algorithm="giac")
Output:
1/2*(b*c - a*d)*arctan(sqrt(d*x^4 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2* c + a*b*d)*b) + 1/2*sqrt(d*x^4 + c)/b
Time = 3.37 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.77 \[ \int \frac {x^3 \sqrt {c+d x^4}}{a+b x^4} \, dx=\frac {\sqrt {d\,x^4+c}}{2\,b}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^4+c}}{\sqrt {a\,d-b\,c}}\right )\,\sqrt {a\,d-b\,c}}{2\,b^{3/2}} \] Input:
int((x^3*(c + d*x^4)^(1/2))/(a + b*x^4),x)
Output:
(c + d*x^4)^(1/2)/(2*b) - (atan((b^(1/2)*(c + d*x^4)^(1/2))/(a*d - b*c)^(1 /2))*(a*d - b*c)^(1/2))/(2*b^(3/2))
Time = 0.24 (sec) , antiderivative size = 218, normalized size of antiderivative = 3.11 \[ \int \frac {x^3 \sqrt {c+d x^4}}{a+b x^4} \, dx=\frac {-\sqrt {b}\, \sqrt {d \,x^{4}+c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d}\, \sqrt {b}\, \sqrt {d \,x^{4}+c}\, x^{2}+\sqrt {b}\, c +\sqrt {b}\, d \,x^{4}}{\sqrt {d \,x^{4}+c}\, \sqrt {a d -b c}+\sqrt {d}\, \sqrt {a d -b c}\, x^{2}}\right )-\sqrt {d}\, \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d}\, \sqrt {b}\, \sqrt {d \,x^{4}+c}\, x^{2}+\sqrt {b}\, c +\sqrt {b}\, d \,x^{4}}{\sqrt {d \,x^{4}+c}\, \sqrt {a d -b c}+\sqrt {d}\, \sqrt {a d -b c}\, x^{2}}\right ) x^{2}+\sqrt {d}\, \sqrt {d \,x^{4}+c}\, b \,x^{2}+b c +b d \,x^{4}}{2 b^{2} \left (\sqrt {d \,x^{4}+c}+\sqrt {d}\, x^{2}\right )} \] Input:
int(x^3*(d*x^4+c)^(1/2)/(b*x^4+a),x)
Output:
( - sqrt(b)*sqrt(c + d*x**4)*sqrt(a*d - b*c)*atan((sqrt(d)*sqrt(b)*sqrt(c + d*x**4)*x**2 + sqrt(b)*c + sqrt(b)*d*x**4)/(sqrt(c + d*x**4)*sqrt(a*d - b*c) + sqrt(d)*sqrt(a*d - b*c)*x**2)) - sqrt(d)*sqrt(b)*sqrt(a*d - b*c)*at an((sqrt(d)*sqrt(b)*sqrt(c + d*x**4)*x**2 + sqrt(b)*c + sqrt(b)*d*x**4)/(s qrt(c + d*x**4)*sqrt(a*d - b*c) + sqrt(d)*sqrt(a*d - b*c)*x**2))*x**2 + sq rt(d)*sqrt(c + d*x**4)*b*x**2 + b*c + b*d*x**4)/(2*b**2*(sqrt(c + d*x**4) + sqrt(d)*x**2))